Generalized indiscernibles from ultrafilters Alex Kruckman March 17, 2021 1 Generalized indiscernibles We fix a complete L-theory T with monster model M and another (unrelated, but arbitrary) language L0. Our goal in this note is to develop some machinery to produce generalized indiscernibles in M indexed by L0-structures. Throughout, let I be an L0-structure which is uniformly locally finite, and let K = Age(I). Note that K is a class of finite L0-structures with the hereditary property (HP) and the joint embedding property (JEP). We will fix some notation. Given L0-structures A and B, we write Emb(A; B) for the set of embeddings A,! B. We will often deal with L-formula or types in variable context (xc)c2C , where C 2 K. Here we have one tuple of free variables xc for each element of C. Given an L-formula '((xc)c2C ) and a C-indexed family of tuples (ac)c2C from M, we can make sense of M j= '((ac)c2C ) in the obvious way. Definition 1.1. Let I = (ai)i2I be a family of tuples from M, indexed by the L0-structure I, and let D ⊆ M be a small set. We say that I is a family of I-indexed indiscernibles over D if for any C 2 K and any embeddings f; g 2 Emb(C; I), we have tpL((af(c))c2C =D) = tpL((ag(c))c2C =D). The behavior of a family of I-indexed indiscernibles over D is completely determined by a family of complete types over D, one for each structure in K. Definition 1.2. A K-type (over D ⊆ M) is a family (pC )C2K, where each pC is a partial L-type over D in variable context (xc)c2C . 0 Given a K-type (pC )C2K and an L -structure J with Age(J) ⊆ K, we define [ [ pJ ((xj)j2J ) = pC ((xf(c))c2C ): C2K f2Emb(C;J) If J = (aj)j2J realizes pJ , we say that J is a J-indexed realization of (pC )C2K. 0 We say a K-type (pC )C2K is consistent if pJ is consistent for every L - structure J with Age(J) ⊆ K. We say (pC )C2K is complete if it is consistent and pC is a complete type over D for all C 2 K. 1 0 If Age(J) ⊆ K, then for any finite tuple j = (j1; : : : ; jn) from J, the L - substructure of J generated by j (call it A) is in K. If the K-type (pC )C2K is complete, then pJ ((xj)j2J ) contains the complete type pA((xa)a2A), which in particular contains a complete type in the variables xj1 ; : : : ; xjn . It follows that pJ ((xj)j2J ) is a complete type in the variables (xj)j2J . Observe that if J = (bj)j2J is a J-indexed realization of a complete K-type (pC )C2K over D, then J is a family of J-indexed indiscernibles over D. Indeed, for any C 2 K and any embeddings f; g 2 Emb(C; J), (bf(c))c2C and (bg(c))c2C both realize the complete type pC over D. So if we want to find generalized indiscernibles, we would like to find (com- plete) K-types. One way to do this is to read them off from a (not necessarily indiscernible) I-indexed family of tuples I = (ai)i2I from M: an L(D)-formula '((xc)c2C ) goes in the type if and only if the set of embeddings f 2 Emb(C; I) such that M j= '((af(c))c2C ) is \large". Of course, we have to decide what we mean by \large". For now, we encode the \large" sets as an arbitrary family G. Definition 1.3. Let G = (GC )C2K be a family of sets with GA ⊆ P(Emb(A; I)) for each A 2 K. Let I = (ai)i2I be a family of tuples from M, and let D ⊆ M be a small set. The G-Ehrenfeucht-Mostowski type of I over D, denoted EMG(I=D), is the K-type (pC )C2K defined by pC = f'((xc)c2C ) 2 L(D) j ff 2 Emb(C; I) j M j= '((xf(c))c2C )g 2 GC g: Example 1.4. The classical EM-type over D of a sequence I = (an)n2! is EMG(I=D), where I is indexed by I = (!; ≤) and G = (GC )C2K, where GC = fEmb(C; I)g for all C 2 K. That is, the only set of embeddings which is \large" is the set of all embeddings. Of course, for an arbitrary set G and an arbitrary family I = (ai)i2I , the K- type EMG(I=D) will typically not be complete (or even consistent). To ensure completeness, we need the sets GC in G to be ultrafilters on Emb(C; I) for all C 2 K, and to ensure consistency, we need these ultrafilters to cohere in a precise sense. 2 Age ultrafilter families Given A; B 2 K and an embedding f 2 Emb(A; B), we obtain a \restriction" map (− ◦ f): Emb(B; I) ! Emb(A; I). Recall that given a filter F on a X and a function ρ: X ! Y , we can \push forward" F along ρ, obtaining a filter ρ∗F on Y , defined by −1 ρ∗F = fZ ⊆ Y j ρ [Z] 2 F g: We write βX for the Stone space of ultrafilters on X. If U is an ultrafilter on X, then ρ∗U is an ultrafilter on Y , so ρ∗ is a map βX ! βY . In particular, for every embedding f 2 Emb(A; B), we obtain a \push for- ward along the restriction" map (−◦f)∗ which maps (ultra)filters on Emb(B; I) to (ultra)filters on Emb(A; I). 2 Definition 2.1. An age filter family on I is a family F = (FA)A2K such that: (1) For each A 2 K, FA is a filter on Emb(A; I). (2) For each embedding f : A,! B with A; B 2 K,(− ◦ f)∗(FB) = FA. We call (2) the pushforward condition. The age filter family F is proper if each FA is a proper filter. It is an age ultrafilter family if each FA is an ultrafilter. 0 0 0 We say an age filter family F = (FA)A2K extends F if FA ⊆ FA for all A 2 K. Lemma 2.2. Let I = (ai)i2I be a family of tuples from M, and let D ⊆ M be a small set. (1) If G extends to a proper age filter family on I, then EMG(I=D) is a consis- tent K-type. (2) If U is an age ultrafilter family on I, then EMU (I=D) is a complete K-type. Proof. For (1), let F = (FC )C2K be an age filter family on I = (ai)i2I extending G. It suffices to show that EMF (I=D) = (pC )C2K is a consistent K-type. Let J be an L0-structure with Age(J) ⊆ K. We would like to show by compactness that pJ ((xj)j2J ) is consistent. A finite subset of pJ has the form f'i((xfi(c))c2Ci ) j 1 ≤ i ≤ ng, where for each i, Ci 2 K, fi 2 Emb(Ci;J), and 'i((xc)c2Ci ) 2 pCi . Let B be the Sn substructure of J generated by i=1 fi[Ci]. Then we can view each fi as an embedding Ci ,! B. Let 'bi((xb)b2B) be the formula obtained from 'i by replacing each tuple of variables xc by xfi(c) and adding dummy variables xb for all b 2 B n f[Ci]. It suffices to show that f'bi((xb)b2B) j 1 ≤ i ≤ ng is consistent. Since 'i((xc)c2Ci ) 2 pCi , we have Xi = fg 2 Emb(Ci;I) j M j= 'i((ag(c))c2Ci g 2 FCi ; so the preimage of this set under (− ◦ fi) is in FB. That is: −1 (− ◦ fi) [Xi] = fh 2 Emb(B; I) j M j= 'i((ah(fi(c)))c2Ci )g = fh 2 Emb(B; I) j M j= 'bi((ah(b))b2B)g 2 FB: Now since FB is a proper filter, we can find some embedding h 2 Emb(B; I) Tn −1 Vn with h 2 i=1(− ◦ fi) [Xi]. Then we have M j= i=1 'bi((ah(b))b2B), so this set of formulas is consistent, as desired. For (2), since U = (UC )C2K is an age filter family on I, EMU (I=D) = (pC )C2K is a consistent K-type by (1). It remains to show that each pC is a complete type. So let '((xc)c2C ) be a formula. The sets ff 2 Emb(C; I) j M j= '((af(c))c2C g 3 and ff 2 Emb(C; I) j M j= :'((af(c))c2C g are complementary, so one of them is in UC , and hence either '((xc)c2C ) or :'((xc)c2C ) is in pC . Example 2.3. Let I = (N; ≤). Then K = Age(I) is the class of all finite linear orders. For any linear order A with n elements, the set Emb(A; I) can be identified with the set [N]n of strictly increasing n-tuples from N. So an age ultrafilter family on I is essentially a family (Un)n2N, where Un is an ultrafilter on [N]n, satisfying the pushforward condition. Given such a family U = (Un)n2N, U1 is just an ultrafilter on N, which must be non-principal. Indeed, suppose that for some a 2 N, fag 2 U1.